Daughters 2010 Honda Civic has 150K , maybe less, auto transmission.... She paid a mechanic to replace the motor mounts last year and now when car first starts there is severe vibration noise. Did the mechanic buy cheap mounts off Amazon? Could the noise be something else like the belt tensioner? Did he not replace all the mounts?
If they did indeed buy cheap mounts, replace them all with OEM/very good quality mounts. Maybe even the transmission mounts.
You don’t want to have to repeat the job three times.
First of all, there's a visual test for the engine mounts. One person looks down at the engine (hood up). The other starts the car. If the engine jerks or jumps, back or forth or up an down, then move on to test two.
Test Two: There's a relatively simple test you can perform, with a floor jack and a few short wood pieces, for support. Put the short wood blocks on the jack, and GENTLY lift the oil pan a few inches, with the jack. Then start the car. If the vibration stops, then it's the motor mounts.
So, rather than your daughter guessing, see if she has a competent friend to help with these visual and tactile tests.
If it is the motor mounts, replacement OEM motor mounts only. No Chinesium mounts.
If it's from the rear of the car, the catalytic converter or its shield could be coming apart or is loose. If it's from the front of the car, I'd be looking at the belt drive system, pulleys and tensioners. You would have to check the actual invoice to see what was replaced. The check it visually on the vehicle.
"when car first starts there is severe vibration noise"
Mounts are an unlikely cause of problems occurring at first startup only, imo.
Why not first try to find the source of the vibrations or noise? The (probably one and only) good thing about severe car vibrations and noises is that those are usually relatively easy to locate by using your ears (or some primitive stethoscope). This appears to be a more realistic way of pinpointing noises, than asking the web community.
Not if the car is in a different state and you are not able to look at it or use a stethoscope, so I do ask the web community. It is possible someone else has had a similar problem they could relate, then I could advise daughter what to look at first. Sorry you couldn't help.
Some modern telecom appliances aka mobile phones have software in them which enables remote communication with sound and video. When I had a problem with my fridge, this was enough for the fridge technician to pinpoint the source of a strange noise, and even to do diagnose the fridge deeply enough for him to order the required spare parts (fan motor) without having to visit me in my office. These modern technologies do magic. And (hint) can be applied to cars too 😉
You might even want to have your daughter film the noise with her mobile, place it in some sort of web cloud service (e.g. youtube), and ask Mr. Kilmer to try diagnose the noise root cause based on this footage - this could be both very helpful and fun.
Just asking the web community "what some strange noise might be" based on a verbal noise description is really not a very promising strategy, cause strange noises could have thousands of different reasons. In my case, I spent 2 years pinpointing a start-up noise in my car, till I found a screwdriver the previous owner forgot in the rain duct under the windshield... So my suggestion to first try locate the noise source yourself (despite the fact that your daughter lives far from you - just try using some mobile phone-based video comm service, really) was not meant as an offence / to be dismissive, but was meant as a practical advice based on my own experience in order to make your noise pinpointing process easier, not harder.