It seems plausible to me, and I would like to know how to verify it.
Why I think xahesh might be cognate with question:
xahesh (IPA: /xɑːheʃ/) in Modern Persian is a noun meaning "request, plea". The root xah- is from the verb xastan /xɑːstæn/ meaning 'to want, desire'. It is spelled xwahesh, and was pronounced /xwɑːhesh/, as it still is in some dialects like Dari. The suffix -esh is from Middle Persian -eshn (/eʃn/) (or maybe -ashn) , which has essentially the same function as Latin -tio, tionis, turning a verb into a noun, though I'm not sure if these two are related.
Several words in Persian beginning with xw- appear to be derived from PIE roots beginning with *sw. For instance xwab (/xɑːb/, formerly /xwɑːb/) "sleep" is listed in some etymology dictionaries as a derivative of PIE *swep, cognate with Greek ὕπνος, and Latin somnus. Another example is xwahar (/xɑːhæɾ/ or /xwɑːhæɾ/) "sister", is clearly from PIE *swésōr, cognate to sister, Schwester, soror, etc. This does not appear to be a strict rule however, since for instance xwandan (/xɑːndæn/, /xwɑːndæn/) "to read, sing", is supposedly cognate with Latin canō, canere, from PIE *kan, which is not labialized.
Given that Latin quaero is rhotacized quaeso, which resembles the h/s counterpart of xwah-, and has a similar sense to it, I think it's reasonable to guess that the two may be cognates, but I've not been able to find this written down anywhere.
Now, this is almost certainly because I don't know where to look. I'd appreciate a response pointing out what is the right academic source or procedure for verifying such guesses. For instance, are there standard academic references for looking up the PIE roots of Persian/Iranian words?