How can I get my Rover dog interested in walks ?

First thing I would do is to walk the dog in the yard because you say the dog is comfortable in the yard. The dog may be a stubborn walker because they fear some part of the experience, be it the leash, passing cars, or simply walking around outside their home and they don't know or trust you yet. Desensitization is a lengthy process in which you introduce your dog to graduating levels of the stimuli that they fear. The idea is that as they become comfortable with a low level of the stimuli, you add a bit more until they are again comfortable. Desensitizing your dog may be the key to your dog becoming happy to go on walks.

Also, have some high value treats to reward the dog for cooperating.

Not all dogs are alike. Sometimes dogs don't like walks, and they don't find a formal walk in the neighborhood appealing. If you're being paid to walk the dog, perhaps a conversation with the owner is in mind. Would the owner be amenable to changing the visit to a drop-in and let the dog do what it would like to do? Just an idea.

Treats! Lots and lots of healthy treats (only give a few but use as motivation)!

Depending on the size of the dog, my first advise will be to carry it for a little, so it starts feeling more constable outside, then start by giving him rewards just for small trips to the outside, just a little walk no more than 5 minutes in the back yard, or just just to the sidewalk & back, then increase the length exponentially by making it to the next house, the corner, around the block & so on, until you can complete full walks with your dog

I hope this can help

My dog used to lay down on walks as well. I started doing was making the walk a 'fun' part of the day. I incorporated treats, positive reinforcement and lots of "good boy's. I also noticed that the street we were talking walks in was too busy and noisy. A much quieter street helped and he is more at ease and happy about walks now.