It's a bit tricky, but possibly? Technically if you claim Sick Leave or Family Leave, you should have reported that as income when applying for Unemployment, which likely made you not eligible for Unemployment. If you claim the Sick Leave and/or Family Leave, you should contact your State on how to repay the amount of Unemployment that you were not eligible for.
From the US Department of Labor:
May I collect unemployment insurance benefits for time in which I receive pay for paid sick leave and/or expanded family and medical leave?
No. If your employer provides you paid sick leave or expanded family and medical leave, you are not eligible for unemployment insurance. However, each State has its own unique set of rules; and DOL recently clarified additional flexibility to the States (UIPL 20-10) to extend partial unemployment benefits to workers whose hours or pay have been reduced. Therefore, individuals should contact their State workforce agency or State unemployment insurance office for specific questions about eligibility.
https://www.dol.gov/agencies/whd/pandemic/ffcra-questions#29
