I noticed that quinque ends in -que. I asked my teacher if this was sheer coincidence or so reason for it. He didn't know but he thought it was coincidental. I, however, think that they probably share an etymology root. So is there a relation, if so what is it?
None, and here's why.
If you look at -que cognates (Hittite -kku 'now, even, and'; Sanskrit -ca 'and'; Greek Gr. -τε 'and' etc.), you will see that the meaning is virtually the same (and). There is no evidence to suggest that -que was an enclitic in five. What would it mean, "five.and", anyway??? And there are no other numerals of the type "X.and." (e.g., four.and, ten.and).
If you look at quinque cognates (Skt. páñca 'five', YAv. paṇca 'five', Gr. πέντε, Arm. hing 'five'; Lith. penkì 'five'), you will see that the segment *kwe is present in a number of languages, which mostly likely means it was inherited from PIE - otherwise, you will have to explain how and why it emerged in all those languages (Sanskrit, Greek, Lithuanian, Armenian etc.).
There is no evidence to suggest that *kwe in five was an enclitic.
Usually, grammaticalization works this way: a free (standalone) word may become a "less free" morpheme (i.e. an enclitic or affix), not the other way (although there might be some extremely rare counterexamples).
Here's what de Vaan gives:
-que [cj. postpos.] ‘and’
Proto-Indo-European *-kwe 'and, -ever'.
Cognates Italic Ven. -kve, Fal. -cue 'and', O.U. -p, -p, O. -p]e 'and'. Indo-European Hit. -kku 'now, even, and', Pal. -ku 'and?', CLuw. -ku 'and, furthermore?'; Skt. -ca, Av. -ca, OP -cā 'and, also, if', Gr. -τε 'and', Myc. -qe 'and'; Go. -uh 'and, also', nih 'if not'.
quīnque [num. indecl.] ‘five’
Proto-Italic kwenkwe 'five', kw(e)nkwto- 'fifth'.
Proto-Indo-European *penkwe 'five', *pnkw-(t)o- 'fifth'.
Cognates Italic O. pumperias , púmperias [nom.pl.] , púmperiais [dat.abl.pl.], U. pumpeřias [nom.pl.] in O. probably 'fifth day', in U. unclear meaning, < * pompe '5' + *-(e)ro + i̯o- (er probably from * peteria- << * peturia- '4'); O. pomtis 'five times' < * pompe-tis, suffix *-is (Lat. -iēs) >> *-tis by analogy with sept-, oct- cent-. U. puntes [nom.pl.], puntis [dat.abl.pl.] '?' < * pomp-ti- 'group of five'?
Indo-European OIr. cóic, W. pymp, Skt. páñca 'five', pañcamá- 'fifth', pakthá- PN, paṅktí- [f.] 'set of five', YAv. paṇca 'five', puxδa- 'fifth', Gr. πέντε, Thess., Lesb. πέμπε 'five', πέμπτος 'fifth', Arm. hing 'five'; Alb. pesë, OCS pętь, Ru. pjat' 'five', OCS pętъ, Ru. pjátyj 'fifth', Lith. penkì 'five', peñktas 'fifth', Go. fimf 'five', OHG fimfto 'fifth', ToA päñ, ToB piś ' five'.
There doesn't seem to be any conexion in Proto-Italic, according to the etymological dictionaries. As to whether there is a conexion in Proto-Indo-European, in which both words originate, that is uncertain. Walde-Hoffmann (1938) on quinque offer a suggestion based on Cuny:
Weitere Analyse von quinque unsicher; vergleiche Cuny, Ét[ymologie] prégr[eque] 1 und die folgende (generalisierendes *que wie in lateinischen quisque und in *que-tuor-es ("vier"), sehe quattuor); ...
So perhaps the -que in quinque is related to the generalising que in quisque "every one, whoever" and in *quetuores "four".