If I have postpartum fatigue, could I have thyroid problems?

No. Post-partum fatigue is normal- in fact, it would be unusual if you weren't tired. You just went through labor and delivery and now have a new baby making it difficult to get much sleep, making this a much more likely cause of fatigue than thyroid issues. A blood test can tell for sure if you're worried or have a history of thyroid problems.
Possible. Fatigue is common post-partum due to lack of sleep and new demands but post-partum thyroiditis if fairly common and can lead to hypothyroidism or underactive thyroid so if you are concerned ask your doctor to be tested.
Fatigue common. Fatigue following childbirth is expected. However, postpartum thyroiditis is pretty common, too. If you think you may have thyroid problems, please discuss with your OB or primary care doctor. It's easy enough to get a blood test to check.