The word 'joy' enters English from Old French 'joie,' from Latin 'gaudia' (joys, the plural of 'gaudium'), from the verb 'gaudēre' (to rejoice, to be glad), from PIE *geh₂w- (to rejoice). The phonological transformation from Latin /g/ to French /dʒ/ (and then English /dʒ/) is a regular feature of the Gallo-Romance sound changes that shaped French from Latin — the same shift that produced 'jardin' from 'hortus' (via a Germanic intermediary) and 'genre' from 'genus.'
The compound 'enjoy' (en- + joy, 'to put into a state of joy') and the verb 'rejoice' (re- + Old French 'joir,' to be glad, 'to joy again') both derive from the same root. 'Gaudy' may also belong to this family — it originally meant 'festive' or 'joyfully ornamented' (from Latin 'gaudium') before deteriorating to mean 'tastelessly bright.'
The Latin 'gaudēre' appears in the opening of the Catholic hymn 'Gaudeamus igitur' ('Let us therefore rejoice'), sung at European university ceremonies since the medieval period. The imperative 'gaude' (rejoice!) is preserved in the English phrase 'to make a gaudy' (to hold a festive celebration), still used at some Oxford colleges.