Up until 2011, University of Houston was a depository for Texas state government documents. Though this program no longer exists in its same form, we still maintain our collection of Texas government information. Physical documents can be found in the basement of MD Anderson Library next to the US government documents collection.
For more recent government information, the state of Texas makes a lot of it accessible online. The following are a few highlighted resources to get you started. Much like federal government documents, it can be helpful to consider which specific agency or department within the state government would be most likely to have the information you're looking for and go directly to their website.
Depending on the geographical focus of your research, you might find useful to look at government information from the City of Houston and Harris County.