Above is a drug interaction checker at Drugs.com, you can run your med list through to check for interactions. It is always a good idea to check with your pharmacist and LLMD before starting any new meds.
If doxy is the only med that you are on, Levaquin should not interact negatively with it. That combo is actually used to treat Bartonella and other intracellular infections along with Lyme.
Always best to check with your doc and pharmacist before starting a new medicine when you are already taking a couple other medicines. Best to be safe.