There is any relation between the English verb "ask" and the French expression "est-ce que"?
This is what the Oxford English Dictionary (OED.com) has to say about the etymology of ask:
Cognate with Old Frisian āskia, āschia to demand, to claim (East Frisian easkje), Middle Dutch eiscen, eescen, eischen, eeschen (also (rare) in forms with initial h-) to ask, request, desire (Dutch eischen), Old Saxon ēskon, ēscan to ask, demand, to search for, investigate (also ēscian (rare) to claim) (Middle Low German ēschen , eischen , also (rare) hēschen, heischen), Old High German eiscōn, eiskōn to demand, ask, to search for, look for (Middle High German eischen , heischen, German heischen) < the same Indo-European base as Sanskrit iccháti seeks, wishes, éṣati seeks, Avestan isaiti longs for, Old Church Slavonic iskati to seek, search, Lithuanian ieškóti to search for, and perhaps further with classical Latin aeruscāre to go begging, to ask
Ultimately the Proto Indo European base ask comes from is *h2eys- through Germanic while "est-ce que" is three words: être from PIE /*h1es-, ce from PIE *ḱe (through Latin ecce), and que from PIE *kʷih2 also through Latin. These are not the same roots.