Hmm. There are ways to use crocheting or knitting to unite cloth units, but in all cases, at the very least you will need to have a secure edge - a hem - on the cloth units before starting. Otherwise the knitting/crochet will pull apart the fabric. If you don't sew at all, that could be a problem. But if you can at least manage a firm hem, you can use embroidery blanket stitch, slip stitch crochet, or picking up stitches in knitting to pull a thin yarn or heavy thread through say every second hole in the Aida. Then you can work crochet or knitting as wide/deep as you like to make an edging all the way around your fabric, then unite the two pieces with any of the techniques you would use to unite knitted or crocheted motifs.
I did a jacket like this once, combining strips of fur rescued from a thrift-store faux fur coat with big chunky cable strips of knitting. I used a crochet hook (and a hole making awl) to edge my fur strip with slip stitch, then joined my knitted cables to the now edged and tamed fur strips, using the crocheted chain as my foundation.
Hope this helps, -K.