First of all what does since start of relationship mean? Maybe a more accurate time frame might be more helpful (aka weeks/months)
Also not using protection doesn't necessarily mean you are actively trying aka are you aware if you have had sex during your fertile periods (10-16 days after your period)
If you don't want to get pregnant I would advice contraception. I wouldn't bother with fertility testing if you don't want to actively get pregnant.
If you don't want to get pregnant though I wouldn't tempt faith aka it hasn't happened yet so might be ok... As having children is a pretty big deal, not to be taken lightly
As re contraception you can use 1. Non latex condoms
2. Different type of combined pill/Progesterone only pill/injection/ implant