Bought a home where main bathroom had been renovated. After about a year we started to hear gurgling in the bathtub when the toilet was flushed. I have never heard this before. Could this be incorrect plumbing?

Usually indicates a partial blockage or blocked vent. With both cases you are hearing the pipes venting through the tub. Does the toilet flush properly?

Yes! We put a water saver push button flush on it and it works just fine. (two options)

Vents work 2 ways, they allow air in and out. You flush the toilet and a big slug of water is pushed out. If a vent is plugged, the air is pushed into the tub drain and it tries to balance itself in the p trap. As the water passes the vent, it create a negative pressure and sucks air through the p trap. That is why its important to have an experienced plumber install the DWV system and why codes tell you where fixtures can be placed. In the case of sucking.through a vent, it can empty a p trap and allow sewer gas into the house. I would have someone rod or flush the vents, they get clogged with leaves and acorns, twigs and such. If you have a lot of trees over the vents it's a good idea to install vented cover caps.