Interesting question! I'd say the answer is closer to parallel evolution, with considerable influence in both directions. Remember, the parchment codex became the primary form of book production well before the rise of Islam and it is in the fifth, sixth, and seventh centuries that we see the first generation of illuminated manuscripts-the Vatican Vergil was painted in around 400 CE, the Cotton Genesis in the 5th century-ish, the Vienna Genesis in the 6th, and the Rabbula Gospels in the 6th century all come to mind as extremely canonical examples of pre-Islamic manuscript painting. Byzantine manuscript painting was a fairly important influence on early Islamic art; although relatively few early Islamic manuscript paintings survive we know medieval Muslim artists copied for example at least one very well-known 6th century Greek scientific manuscripts(or an extremely similar manuscript therof) now in vienna because we have a medieval Arabic translation of that manuscript with very similar paintings. On a more calligraphy-oriented note, one fairly likely influence is the practice of dying parchment. Dyed purple parchment was used in late Antiquity before the rise of Islam for luxury manuscripts and it's quite possible that at least one famous Qu'ran manuscript, the so-called Blue Qu'ran, was dyed to echo this practice. Likewise, we know that medieval Arabic-speaking Christians often painted manuscripts in a distinctly Islamicate style.