If you read the full text of the above link you would read the following, however this pertains to only FL, and may or may not be the case while traveling through other states enroute:
QUESTION 6. I am planning a trip to Florida shortly. I do not have a permit from my home state nor do I want to obtain a Florida permit. However, I would still like to have a weapon with me for self-protection. What are my options?
Florida law does allow a citizen to transport a weapon in a private vehicle, even if that citizen DOES NOT HAVE a concealed weapon license. Note the following two key provisions in the law:
Section 790.25(5), which deals specifically with possession in a private conveyance states that "it is lawful and is not a violation of s. 790.01 for a person 18 years of age or older to possess a concealed firearm or other weapon for self-defense or other lawful purpose within the interior of a private conveyance, without a license, if the firearm or other weapon is securely encased or is otherwise not readily accessible for immediate use. Nothing herein contained prohibits the carrying of a legal firearm other than a handgun anywhere in a private conveyance when such firearm is being carried for a lawful use. Nothing herein contained shall be construed to authorize the carrying of a concealed firearm or other weapon on the person. This subsection shall be liberally construed in favor of the lawful use, ownership, and possession of firearms and other weapons, including lawful self-defense as provided in s. 776.012." (Emphasis added.)
Section 790.001(17) defines the term "securely encased" to mean "in a glove compartment, whether or not locked; snapped in a holster; in a gun case, whether or not locked; in a zippered gun case; or in a closed box or container which requires a lid or cover to be opened for access."
So, while you cannot carry the weapon on your person, you can at least have it nearby in your vehicle while traveling.