I'm fairly new to machine frame quilting, but I would say no. The quilt has to be able to fit along the length of the rails. That being said, you should measure the width of your king and the width of the queen frame rails, remembering that you need a couple of extra inches on each side for backing and batting to allow for the quilting "taking up " some of the pieced top. (make sense?) Also, you would have to be careful how much room you have in the harp of your machine. A king quilt is very large and when you roll the take-up rail, the area that your machine has to sew gets smaller and smaller. Also, I turn my quilts to do the borders. If your king is rectangular, you probably wouldn't be able to get it lengthwise onto the rail for sure. Sorry. Maybe a more experienced machine quilter will have better news than I do.
Where are you on Vancouver Island? I moved to Revelstoke 30 years ago from Nanaimo and still want to go back! But I have a husband who loves the mountains.