No, that is not technically true.
[From article 1, section 4 of the Texas Constitution], bolding mine:
Sec. 4. RELIGIOUS TESTS. No religious test shall ever be required as
a qualification to any office, or public trust, in this State; nor
shall any one be excluded from holding office on account of his
religious sentiments, provided he acknowledge the existence of a
The Texas constitution gives protection against religious discrimintation for public office, but that protection does not extend to atheists. However that is long way from saying that atheists can't hold office. It means that under the Texas constitution someone 'could' choose to exclude atheists: but in fact that does not happen. One of the reasons that it does not happen is that it would be illegal under the US constitution.
[From article VI of the US constitution]:
The Senators and Representatives before mentioned, and the Members of
the several State Legislatures, and all executive and judicial
Officers, both of the United States and of the several States, shall
be bound by Oath or Affirmation, to support this Constitution; but no
religious Test shall ever be required as a Qualification to any Office
or public Trust under the United States.
Obviously this clause in the constitution discriminates against atheists - but it is not a ban.