Why do you want to do this? What makes you think your body needs a break from the Synthroid? Our bodies have to have thyroid hormones all the time, not just when we feel like taking the medication.
Because of the long half life of Synthroid (T4 medications), it will take a few weeks for the medication to get completely out of your system, then when you start taking it, it would take another 4-6 weeks for it to reach full potential in your blood again. It's not something we can start and stop at will.
All of that said, can you please post your current lab values so we can see what your actual thyroid hormone status is? Please include reference ranges, as those vary from lab to lab and have to come from your own report.
I'm not sure if you're aware, but hypothyroidism can cause depression/anxiety, so if your thyroid dosage isn't high enough, it could cause you to need a higher dose of the Zoloft. Hypothyroidism, also, causes hair loss, so it sounds as though you may need a higher dose of thyroid medication, not a lower dose or to stop it.
What dose of Synthroid are you on, how long have you been taking it and what other symptoms do you have?