I would say, no.
The Castle Doctrine only deals with civil immunity and a duty to retreat; it does not deal with carrying a weapon.
The Motorist Protection Act (HB 1815) allows anyone to carry a handgun in their vehicle with these exceptions:
"(a-1) A person commits an offense if the person
intentionally, knowingly, or recklessly carries on or about his or
her person a handgun in a motor vehicle that is owned by the person
or under the person's control at any time in which:
(1) the handgun is in plain view; or
(2) the person is:
(A) engaged in criminal activity, other than a
Class C misdemeanor that is a violation of a law or ordinance
regulating traffic;
(B) prohibited by law from possessing a firearm;
or
(C) a member of a criminal street gang, as
defined by Section 71.01."
So, if your weapon is in plain view, (like, when a police officer walks up to the window) I'd say you need to keep it concealed.