Is it permissible to specify a certain number—such as three 'na'ams' (yeses)—for acting upon the Holy Quran, and to restrict oneself to that?

The fundamental principle in worship is strict adherence to revealed texts (Tawqif). Specifying an act of worship with a certain number for which there is no textual evidence is considered an additive innovation (bid‘ah idafiyah). However, what is mentioned in the question, such as repeating a verse three times or recalling three blessings, does not fall under innovation, because the intention behind it is not to worship by a specific number, but rather for reflection, contemplation, and drawing benefit. This is permissible as long as it is for the purpose of remembrance, study, or learning.