Which is grammatically correct? Open or opens?

Solution 1:

Neither is correct. "The quest of finding methods of expression that are authentic to oneself opens up doors" would be grammatical.

"Quest" is a singular noun: "The quest opens up doors". "Methods" is a plural noun: "Methods that are authentic".

The other answer suggests using "The quest of finding methods of [expression that is authentic to oneself] opens up doors", where "that is authentic to oneself" is taken as a relative clause modifying "expression". That is technically a possible structure for the sentence, and the use of "is authentic..." is correct for that structure, but it doesn't sound natural to me at all.