The first question is:
Is there any biblical justification for referring to God as Mother?
It is clear that in the cultures in which the Bible was written, God was overwhelmingly seen as male. Almost all of the language referring to God in the Bible addresses and refers to God as a male. And for Christians, there is the additional fact that Jesus Christ, who is "God with us," was a male human being.
For people in the cultures of the Bible writers, in which women were almost universally seen as inferior to men, to picture God as anything other than a male would have been to demote God to inferior status. (I'm not saying this is the reality; only that this is how it would have been viewed in those cultures.) So finding a biblical justification for referring to God as Mother will be slim pickings at best.
And yet, looking at the Bible from the perspective of present-day Western society, which has moved toward a position of considering men and women equal, the almost universal practice in the Bible of referring to God as a male becomes problematic. It implies that men are by nature superior to women, since God is male and not female.
This is the primary driver behind the widespread contemporary movement in liberal Christianity toward recognizing a female aspect of God, and referring to God not only as Father but also as Mother.
Can it be justified from the Bible?
Not very easily.
However, here are a few biblical indications not mentioned in the other answers so far:
So God created humankind in his image, in the image of God he created
him; male and female he created them. (Genesis 1:27)
Here the Bible says that both male and female humans are created in the image of God. This implies that both male and female characteristics are reflections of the nature of God.
The word here translated "humankind" is adam, which primarily refers to humanity generally (including both male and female human beings), though it can also be used to refer to a male human being. (The specific word for a male human being in Hebrew is ish.) Adam is, however, grammatically masculine, which causes it to take masculine adjectives, pronouns, and so on--hence the reading of "he created him," even though it includes both male and female humans.
For thus says the Lord . . . . As a mother comforts her child, so I
will comfort you; you shall be comforted in Jerusalem. (Isaiah 66:12,
Here YHVH (translated as "the Lord"), the God of the Hebrews, generally seen by Christians as God the Father, compares himself to a mother comforting a child. This suggests that God is not averse to being seen as a mother by us humans, who are God's children.
And one more passage in which God is seen as being like both a master (male) and a mistress (female):
As the eyes of servants look to the hand of their master, as the eyes
of a maid to the hand of her mistress, so our eyes look to the Lord
our God, until he has mercy upon us. (Psalm 123:2)
The philosophical response that "God has no gender" is foreign to the biblical text, which almost everywhere uses gendered language for God. The vast majority of that gendered language is male; a small minority of it is female. I'm not aware of any passage in the Bible stating or implying that God has no gender. If the intention is to move away from the idea of God as exclusively male, it is easier to support from the Bible the idea that God encompasses both genders than it is to support the idea that God has no gender.
The second question is:
Are there any denominations or movements which do call the first
person of the trinity 'Mother'?
As asked, this question excludes my denomination, the Swedenborgian Church of North America, since we do not accept the doctrine of a Trinity of Persons in God. (See my article, "Who is God? Who is Jesus Christ? What about that Holy Spirit?")
However, if the question is broadened to ask, "Are there any denominations or movements which do call God 'Mother'?" I can respond from my denominational perspective.
Among Swedenborgians or "New Church" people historically, God has been seen primarily as male, as in other Christian denominations. However, doctrinally speaking, our church recognizes God not only as human, but as encompassing and having both male and female attributes, and as being the source of both male and female genders and characteristics.
Perhaps the clearest statement to this effect in the foundational doctrinal writings of our church is found in Sacred Scripture #67, by Emanuel Swedenborg (1688–1772), in which he is commenting on the commandment, "Honor your father and your mother":
People think of "father and mother" as their father and mother on
earth, and all who serve as parent figures. They understand honoring
father and mother to mean respecting and obeying them.
Spiritual angels think of God as their father and the church as their
mother. They understand honoring them to mean loving them.
Heavenly angels think of God’s divine love as their father, and God’s
divine wisdom as their mother. They understand honoring them to mean
doing good things from God.
By "people" here Swedenborg means people living on earth. By "spiritual angels" he means angels of the second, or middle heaven. By "heavenly angels" he means angels of the third, or highest heaven (mentioned by Paul in 2 Corinthians 12:2).
In other words, according to Swedenborg, the highest angels, who are the wisest and the closest to God of all the angels, think of God's love as Father, and God's wisdom as Mother. This reflects Swedenborg's overall teaching that male and female, and the relationship between them, come from the love and wisdom in God--which Swedenborg sees as the core reality and being of God. (And note that he here reverses the usual assignment of love to female and wisdom to male--though at other times he sticks with the more traditional assignments.)
On this doctrinal basis, some contemporary Swedenborgians feel very comfortable addressing God as both Mother and Father.
If you are interested in a fuller presentation of some of these ideas from a Swedenborgian Christian perspective, I take it up in a sermon I preached some years ago, which I've now put online here:
"The Mother of All the Living"
My congregation at the time was rather conservative. This sermon therefore created quite a stir among some of the older and more traditional members. Afterwards, one of the patriarchs of the church told me in no uncertain terms that he never again wanted to hear a sermon like that in his church! And yet, to some others in the denomination, that old sermon of mine seems quite tame. I say this to let you know that not all Swedenborgians are comfortable addressing God as Mother.