According to the Oxford English Dictionary, the etymology for trumpet is:
Etymology: < French trompette (14th cent.), dim. < trompe , trump n.1
In addition, EtymOnline adds:
c.1300, from O.Fr. trompette "trumpet," dim. of trompe (see trump (n.2)). The verb is recorded from 1520s; figurative sense of "to proclaim, extol" is attested from 1580s.
The etymology for triumphant, on the other hand, is:
Etymology: < Latin triumphānt-em , present participle of triumphāre to triumph v., or < French †triumphant , triomphant (15th cent.): see -ant suffix.
For this, EtymOnline adds:
late 15c., from L. triumphantem, prp. of triumphare (see triumph).
This does not suggest a direct relation between triumph and triumphant, as the words they can be traced back to differ. Perhaps the French trompette is indeed related to the Latin triumphant (thus giving an older and shared French root), but the OED provides no further history. So, because the two words can be traced back to different French roots, they have different etymologies in English. This does not mean that trompette and triumphant don't share a root in French, but as far as English roots go they are different in this regard.