Was buying no name brand gas ruining my car?

2009 Toyota Corolla le, 1.8 L

I’ve been feeding my car a cheaper gas from a local gas station in my area and it would always make my car an engine vibrate horribly. I thought it was possibly my engine mounts acting up but I took my car to chevron this whole month and I no longer get vibrations at all. It would vibrate so bad in a cold start I would warm it up for 1-3 minutes to get it to stop. No premium gas just regular 87