Both of these buildings have the address of 607 Texas Avenue, thus they'll be covered at the same time. The restaurant at the corner of Texas and Live Oak was built in 1980 (Brazos CAD says 1978 for some reason) with La Quinta Inn built behind it in 1979 (originally "La Quinta Motor Inn", later "La Quinta Inn", before branded as simply "La Quinta") right behind it. The original restaurant was Julie's Place (#139), operated by Associated Hosts Inc. of California, which ran several hotel-affiliated restaurants. Boasting a large menu including hamburgers and onion soup, Julie's Place became notorious after a January 1987 murder (there was a story on MyBCS, though I'm sure I had heard it elsewhere about how the manager actually swallowed the key to the safe and the stabbings were to retrieve the key, but I'm not sure on that since that's just a comment on the forum and the official court summary makes no mention of the key-swallowing incident). That said, an article from the Houston Chronicle did mention the body was cut from the sternum to the pelvis, which lends credence to the statement.
In May 1988 Associated Hosts closed the restaurant and reopened it as Bombay Bicycle Club a month later, but in 1995 it closed permanently and became a Denny's in 1996.
Additionally, the La Quinta has some additional buildings (which appear to have been shuttered) behind what used to be Rice Garden and the La Quinta Inn was previously home to a "super slide" of some sort, but I can't find much information on that. (Parts of this post originally appeared here).
UPDATE 09-13-2024: Update with new opening dates and links.