In the scriptures we do not see the lord falling sick or having a fever. He was tired, was thirsty, hungry and he was fatigued. Could he have fallen sick? Or was he immune to any and all sicknesses?
Could Jesus have fallen sick?
There are only two possible answers: "yes" or "no".
The argument for "yes" would be based on Jesus being in all ways fully man, and fully "tempted in every way such as we are, yet without sin" Hebrews 4:15.
And by His perfectly fulfilling the requirements of being the Passover Lamb for all His people, by having "no blemish" Exodus 12:5.
Whether or not having "fallen sick" (and what kinds of illness might be in the [dis]qualifying list) would equate to not being "without blemish" Hebrews 9:13-14, is open for debate, I suspect.
The answer for "no" can really only be supported by the lack of record of Him ever being ill.
What the bible is silent on, and cannot be fully logically deduced from what it isn't silent on, has to be left to the realm of "we don't know".
The Bible informs us that Jesus met with the outcasts of society and healed people with leprosy (Matthew 11:5; Luke 17:12). There is no record anywhere in the Bible that Jesus succumbed to sickness or disease. Yes, he felt fatigue, thirst, hunger and ordinary human emotions – the foremost being compassion – yet not a word do we hear of Jesus becoming sick. What the Bible does tell us is that Jesus was without sin:
“You know that he appeared in order to take away sins, and in him there is no sin” (1 John 3:5).
“He committed no sin and no deceit was found in His mouth” (1 Peter 2:22).
He was “a lamb without blemish or defect” (1 Peter 1:18-19).
John 1:1 says “the Word was God.” John 1:14 says “the Word became flesh.” Jesus is the Word. This clearly indicates that Jesus is God in the flesh – fully God and fully man.
Jesus was the eternal Son of God incarnate. Wicked men could kill His body, but they could not change His eternal nature or diminish His divine power. In John 10:17–18 Jesus says something that no mere mortal could ever say: “I lay down my life—only to take it up again. No one takes it from me, but I lay it down of my own accord. I have authority to lay it down and authority to take it up again.” No one else in the history of the world has ever had the authority both to lay down his life and to raise it up again. Jesus was no mere, sinful mortal, destined to become old and sick and die.
As Nigel J has pointed out, yours is a hypothetical question. Nonetheless, it is profitable for us to understand who Jesus Christ REALLY is.