The ATLAS trigger system is essential to efficiently select the events of high interest for physics analyses. Development of a new muon trigger is ongoing for High-Luminosity LHC, which is scheduled to start in 2026. The first stage of the new trigger system assumes muon track reconstruction from Thin Gap Chamber (TGC) hits with an angular resolution of $4~\rm{mrad}$. An algorithm with pattern matching for a pseudorapidity range of $2.13$--$2.16$ was implemented in an XCVU9P FPGA and demonstrated with test samples including seven hits on seven TGC layers. The implemented algorithm reconstructed tracks successfully with an angular resolution of less than $4~\rm{mrad}$. In addition, GTY transceiver on XCVU9P FPGA was tested. Measured bit error ratio was less than $2.5 \times 10^{-16}$, and the power consumption for 100 pairs of transmitters and receivers was estimated to be about $30~\rm{W}$.