If I understand you correctly you are stating that the Quran makes the following claim: God cannot be the Father of Christians because He allows bad things into their lives and if He was a loving Father He would not allow them to suffer. You would like to know how to falsify this claim.
This is a large topic but is fairly well falsified in the Bible. The first principle to hold onto is that all suffering, whether sourced in the natural world (disease, catastrophe, etc.) or in the actions of men, finds its ultimate cause in the Garden of Eden. When Adam partook of what had been prohibited, all of creation was placed in bondage to corruption:
For I consider that the sufferings of this present time are not worth comparing with the glory that is to be revealed to us. For the creation waits with eager longing for the revealing of the sons of God. For the creation was subjected to futility, not willingly, but because of him who subjected it, in hope that the creation itself will be set free from its bondage to corruption and obtain the freedom of the glory of the children of God. For we know that the whole creation has been groaning together in the pains of childbirth until now. And not only the creation, but we ourselves, who have the firstfruits of the Spirit, groan inwardly as we wait eagerly for adoption as sons, the redemption of our bodies. - Romans 8:18-23
Looking at the account of the fall of man in Genesis 3, we can see hints of the purpose of God in cursing His own natural world:
And to Adam he said, “Because you have listened to the voice of your wife and have eaten of the tree of which I commanded you, ‘You shall not eat of it,’ cursed is the ground for your sake; in pain you shall eat of it all the days of your life; thorns and thistles it shall bring forth for you; and you shall eat the plants of the field.
By the sweat of your face you shall eat bread, till you return to the ground, for out of it you were taken; for you are dust, and to dust you shall return.” - Genesis 3:17-19
Some translations read "because of you" but a clearer understanding is "for your sake". Because Adam sinned while in the condition of perfect provision, perfect health, and unstained fellowship with God, and because all of mankind has inherited the disposition of sin from Adam, it could not be the continuation of those conditions that would bring sinful man to repentance. For the world at large, difficulty of all kinds is allowed by God in order to cause them to repent and cry out to Him. It is not in our nature to seek after God when life is easy:
Two things have I required of thee; deny me them not before I die: Remove far from me vanity and lies: give me neither poverty nor riches; feed me with food convenient for me: Lest I be full, and deny thee, and say, Who is the LORD? or lest I be poor, and steal, and take the name of my God in vain. - Proverbs 30:7-9
Why, then, would God allow suffering in the lives of those who have repented, cried out to Him, and been saved by Him in the Christian sense? If believing in the Lord Jesus Christ makes one into a son of God, why does suffering continue and sometimes even intensify? I would like to offer two main purposes of Christian suffering from Scripture:
The conversion of a person from child of wrath to child of God is both a one time event and a process. One does not need to be born again by the grace of God through faith in Jesus multiple times:
But God, who is rich in mercy, for his great love wherewith he loved us, Even when we were dead in sins, hath quickened us together with Christ, (by grace ye are saved;) And hath raised us up together, and made us sit together in heavenly places in Christ Jesus: That in the ages to come he might shew the exceeding riches of his grace in his kindness toward us through Christ Jesus. For by grace are ye saved through faith; and that not of yourselves: it is the gift of God: Not of works, lest any man should boast. For we are his workmanship, created in Christ Jesus unto good works, which God hath before ordained that we should walk in them. - Ephesians 2:4-10
but one does need to learn how to trust in the Lord in all circumstances in order to become like the perfect Son, Jesus. John 1:12 says that those who have received Jesus are given the power to become the sons of God. This power is further explained in Ephesians as faith given as a gracious gift of God. The culmination of this power of faith exercised in a corrupt world is the redemption of our bodies from the dead, which Scripture calls the adoption:
And not only they, but ourselves also, which have the firstfruits of the Spirit, even we ourselves groan within ourselves, waiting for the adoption, to wit, the redemption of our body. - Romans 8:23
If Jesus, the perfect Son, endured hardship looking for the joy that was set before Him, then we should certainly expect nothing less:
Looking unto Jesus the author and finisher of our faith; who for the joy that was set before him endured the cross, despising the shame, and is set down at the right hand of the throne of God. - Hebrews 12:2
Indeed, we are exhorted to endure hardship as loving Fatherly discipline. Notice how the writer of Hebrews points to the reverence shown by children to earthly fathers by virtue of their discipline and indicates that the reverence due our heavenly Father should be so much more:
or consider him that endured such contradiction of sinners against himself, lest ye be wearied and faint in your minds. Ye have not yet resisted unto blood, striving against sin. And ye have forgotten the exhortation which speaketh unto you as unto children, My son, despise not thou the chastening of the Lord, nor faint when thou art rebuked of him: For whom the Lord loveth he chasteneth, and scourgeth every son whom he receiveth. If ye endure chastening, God dealeth with you as with sons; for what son is he whom the father chasteneth not? But if ye be without chastisement, whereof all are partakers, then are ye bastards, and not sons. Furthermore we have had fathers of our flesh which corrected us, and we gave them reverence: shall we not much rather be in subjection unto the Father of spirits, and live? For they verily for a few days chastened us after their own pleasure; but he for our profit, that we might be partakers of his holiness. Now no chastening for the present seemeth to be joyous, but grievous: nevertheless afterward it yieldeth the peaceable fruit of righteousness unto them which are exercised thereby. - Hebrews 12:3-11
It is important to note here that a direct rebuttal of the Quran's claim (assuming the claim is true) is found in this passage in verse 8. If (theoretically) one is not allowed hardship it stands as proof that one is not a child of God. I say theoretically since everyone, without exception, experiences suffering because God would have everyone become His child:
But if ye be without chastisement, whereof all are partakers, then are ye bastards, and not sons. - Hebrews 12:8
Not all of my suffering is for my benefit. Just as Christ suffered for the sins of many and, by His death many are made righteous, so we are called to endure our sufferings for the benefit of others, not railing against them as we used to do but receiving them as from the hand of a loving Father who is trying to reconcile the whole world unto Himself:
For what glory is it, if, when ye be buffeted for your faults, ye shall take it patiently? but if, when ye do well, and suffer for it, ye take it patiently, this is acceptable with God. For even hereunto were ye called: because Christ also suffered for us, leaving us an example, that ye should follow his steps: Who did no sin, neither was guile found in his mouth: Who, when he was reviled, reviled not again; when he suffered, he threatened not; but committed himself to him that judgeth righteously: Who his own self bare our sins in his own body on the tree, that we, being dead to sins, should live unto righteousness: by whose stripes ye were healed.
For ye were as sheep going astray; but are now returned unto the Shepherd and Bishop of your souls. - 1 Peter 2:20-25
When one perseveres in faith during adversity then one demonstrates to self and others that the Lord's promises to us are true:
These things I have spoken unto you, that in me ye might have peace. In the world ye shall have tribulation: but be of good cheer; I have overcome the world. - John 16:33
For whatsoever is born of God overcometh the world: and this is the victory that overcometh the world, even our faith. Who is he that overcometh the world, but he that believeth that Jesus is the Son of God? - 1 John 5:4-5
and is able to comfort those who are undergoing similar difficulty:
Blessed be God, even the Father of our Lord Jesus Christ, the Father of mercies, and the God of all comfort; Who comforteth us in all our tribulation, that we may be able to comfort them which are in any trouble, by the comfort wherewith we ourselves are comforted of God. For as the sufferings of Christ abound in us, so our consolation also aboundeth by Christ. And whether we be afflicted, it is for your consolation and salvation, which is effectual in the enduring of the same sufferings which we also suffer: or whether we be comforted, it is for your consolation and salvation. And our hope of you is stedfast, knowing, that as ye are partakers of the sufferings, so shall ye be also of the consolation. - 2 Corinthians 1:3-7
There is so much more to say, but in these two points alone, there is a strong rebuttal to the idea that God, as loving Father, should erase all difficulty immediately. He will someday completely eradicate all sin and suffering, but currently He is working in the midst of them to produce and perfect children through Christ Jesus, His Son and our Lord.