Do I have to use an FR4 material with a high Tg (Tg = glass transition temperature) for lead-free soldering?

FAQ about printed circuit boardsMATERIAL

No, not necessarily. There are many factors to be taken into account, e.g. how many layers, the thickness of the PCB and also a good understanding of the assembly process (number of soldering cycles, time above 260 degrees, etc.). Some research has shown that a material with a “standard” Tg value has even performed better than some materials with a higher Tg value. Note that even with “leaded” soldering the Tg value is exceeded.
What is of most importance is how the material behaves at temperatures above the Tg value (post Tg) so knowing the temperature profiles the board will be subjected to will help you look evaluate the necessary performance characteristics.