Texas counties have no general power to require a tree permit on private land, so unincorporated Williamson County has no county tree permit. Deed restrictions, HOA rules and utility easements can still apply. If you are in a city's ETJ, the City of Austin tree ordinance does not apply there.
Texas law: cities cannot charge a tree mitigation fee for a tree under 10" at an existing one- or two-family home you live in (Local Gov't Code §212.905, HB 7, 2017). Since 2023, landowners can petition to leave a city's ETJ (SB 2038), so ETJ lines change; we read the live City of Austin jurisdiction map.
Official sources: statutes.capitol.texas.gov/Docs/LG/htm/LG.240.htm · austintexas.gov/development-services/city-arborist