No, you can become fertile before you have a period, but you are not likely to know about it so easily.
You can check cervical mucus, use temperature charts etc etc but I suggest you chill out for a few months, shag every couple of days and enjoy it.
The reality is that if you don't get pregnant easily then TTC will become very ritualistic and less enjoyable, so enjoy the time you have without all of that.
Also, and here is just a small word of warning. There are a lot of people on these boards who have struggled to become pregnant and have taken years and years to get pregnant, if they have managed it at all. It is really very frustrating when someone who came off the pill 6 weeks ago starts a thread saying, 'am I pregnant?', 'why am I not pregnant?', 'do you think I could be pregnant?'.
Long-term TTC is joyless and frustrating. You may get lucky, you may not, but please spare a thought for those who are stuck in an endless cycle of hoping month on month and hugs disappointment.
Sit tight, wait and see if your period pitches up and shag lots.