As simple as it sounds, it requires mastery of the English language to use whom in the right situation. In fact, if at all possible, always try to rephrase your sentence to use 'who' instead of 'whom'.
In your sentence, unfortunately, even if 'whom' was correct, it would be referring to professors' trust as the object and not the intended porfessors'
The best way of writing this sentence:
"As a result of gaining my professors' trust by submitting assignments accurately, and in a timely manner, my professors granted me more advanced theoretical tasks."
The correct way of writing this sentence using who:
"My professors gained trust in me as a result of my timely and accurate submissions, who then granted me more advanced theoretical tasks."