Front he sounds of it your car is an auto... I am having the exact same issue but mines a stick. At 60 Kph my car starts vibrating quite a bit.... It seems to go away after about 15 minutes of driving. I took it to the dealer twice and they told me it was the tires because I let it sit too long (It's on stands when in storage so I highly doubt that). They also told me if I bring it in again and they don't find anything they would charge me for the visit.

So, I went and installed new tires even though the original tires only had 20000 KMs on them... Same issue. Now my warranty is over and the issue is getting worst. So frustrating.
A second symptom I have is occasionally when I come off the highway the clutch is rock hard and I have to depress the clutch pedal 5-6 times to soften it up.

I'm almost afraid to find out what it is and the cost associated.