Taking the cue words from @Sola Gratia stating the Greek sense of the word "grew in grace" as pertaining to "maturity", we can also relates & infer this interpretation to the Mystery of Finding Jesus in the Temple. Jesus eagerly wanted to do God's Will in redeeming mankind even at the young age of twelve. Imagine Jesus at that age calling the apostles come and follow me.
Gospel of Luke painted Jesus humanity still not matured spiritually as we can see in Luke 2:52. Jesus still needed to grow in wisdom and grace. Jesus needed 18 more years to become fully ready both physically and spiritually as a human being by living a life of submission and obedience to St.Joseph and Mary.
At the wedding in Cana, St.John gospel described Jesus still not ready to start his ministry at 30 years old saying;
"Woman, my hour has not yet come."( John 2:4 )
But Mary this time knew Jesus was already matured to start his saving ministry unlike before when he was only twelve years old. Mary knew it was already the right time. The Wedding at Cana is the 'fullness of time' for Jesus to start his redemptive mission.
And the Woman intercessory power was revealed and her docility to the voice of the Holy Spirit shines through instructing the servants;
"Do whatever he tells you."( John 2:5 )
This particular passage reveals Jesus and Mary's role as the New Adam and New Eve. The six jars represents six days of creation and Jesus promised that He will make all things new. A new creation by restoring Divine Life and offering immortality by giving Eternal Life thru partaking of the Holy Eucharist.
Gospel of John narrated Jesus only received the anointing of the Holy Spirit at the Baptism in the River Jordan. So, from 12 years old to 30 years old, St.Joseph and Mother Mary was helping in preparing Jesus for his redemptive mission.
On the other hand, it was angel Gabriel greetings prior to the Mystery of Annunciation that reveals the "fullness of time" had arrived in Luke gospel;
"Hail, Full of Grace.."( Luke 1:28 )
The "fullness of time" in Galatians 4:4 had arrived at the right time when the handmaid of the Lord had reach the maturity in faith. Her famous Fiat can only be uttered with full trust after she merited abundant graces from living a life of humility and obedience to God's Will. The ten years of services she offered at the Temple of God away from affairs of the world increases her holiness and perfected her virtues according to Eastern Tradition, and they acknowledge Mary as the "All Holy or Panagia".
Angel Gabriel had revealed the beautiful and perfect state of soul of the Blessed Virgin Mary and its now ready to fulfill Her role as the Theotokos. Blessed Virgin Mary was now fully matured in faith and responded with full trust to God's Will in the Divine Plan of Salvation, saying to Angel Gabriel her famous Fiat;
"Be it done unto me according to thy Word.( Luke 1:38 )
The answer is Yes! ,the "fullness of time" depends on Mary's state of grace.
Will God's infinite wisdom entrust the Logos to a corrupted vessel and to an ordinary woman who do not possessed holiness and perfected their virtues?
Divine Motherhood is not a simple task, it emcompases a royal dignity not just on earth but also in heaven. Mary is the cause of Man's salvation according to St.Irenaeus a Church Father.
Without the Fiat of the Blessed Virgin Mary there is No Redemption and there is No Salvation for all sinners.
Jesus needed a pure body and the Holy Spirit needed a sorrowful & immaculate heart and the Abba Father needed a humbled & surrended will.The Most Holy Trinity resided fully in the body,soul and heart of Mary, a perfect and complete union mystically.
Sweet heart of Mary be my salvation, is reflected in Book of Wisdom.
" Who has learned your counsel, unless you have given wisdom and sent your holy spirit on high?
And thus the paths of those on earth was set right, and people were taught what pleases , and were saved by wisdom."(Wisdom9:17-18)
The people are saved by Wisdom and Mary is the Seat of Wisdom.
"No Mary, No Jesus" (St.Mother Theresa of Calcutta)
EDIT: Clarifications on @Sola Gratia comment.
"Scripture doesnt say Jesus had no grace.. "
Scripture really stated Jesus had emptied himself ( Philippians2:7 ), even Jesus does not need "sanctifying grace to begin with". And the child Jesus to merit actual graces must obey and submit to St. Joseph and Blessed Virgin Mary. ( Luke 2:52 )
Jesus does not need sanctfying grace because his role is a Redeemer or a Saviour. But St. Luke specifically stated and described how Jesus humanity acquired actual graces and wisdom:
" Then he went down to Nazareth with them and was obedient to them. "( Luke2:51 )
Jesus humanity by living a life of humble obedience to St. Joseph and the Theotokos had acquired graces in preparation for His Redemptive Mission.