I have found Connecting Threads fabrics will fray before you even wash them so have discontinued ordering from then for that reason and I used to love their fabrics especially their blender and batiks as the prices were right up my alley. As someone has mentioned, check your 1/4" seam and your stitch length, tighten it and that might help you. You'll find some of JoAnne's fabrics had the same problem years ago and so I quit buying from them too. I don't have a JoAnne's neare me anymore so that's helped me.

I would recommend starching your fabric before cutting. It makes makes
a big difference. Not only with fraying but it will help with your piecing, etc.
Of course it's always good to check that your pieces are properly aligned.
When you are pinning your blocks together, check the other side before
sewing. Happy sewing!

Also check to be sure your fabrics are lined up before you stitch your 1/4 seam. Sometimes the bottom fabric slides off. I don’t prewash or starch and press my seams open and don’t have that problem
unless I don’t notice the two fabrics aren’t lined up.

good advice given so far.....make sure you have a good 1/4 inch seam.....I generally check that I have that when I press my seams and I usually press to one side....sometimes it is easy to get a 1/4 inch on the fabric piece that is on top...but the bottom fabric moves a bit and maybe only a bit of it gets caught in the seam....so check that bottom side too for seam width. A shorter stitch length also helps....I general move mine down from the default of 2.5 to 1.8 or 2.0 depending on how small my pieces of fabric all. A shorter stitch helps keep the seams closed but is harder to rip out to fix errors....not that I make many...just kidding...my seam ripper and I know each other well. I usually stay stich about 1/8 of an inch around the whole quilt top when done as it helps keep seams from becoming undone while handling the top preparing to sandwich and quilt it. I sometimes also do that around blocks if there are many pieces in the block and I will be putting them out to arrange the blocks..again, more handling of those fragile seams. also, I chain piece when piecing but try and do a couple of stitches between feeding the pieces in...that way I can cut in the middle of those stiches leaving a little bit extra thread at the edge of the block and it seems to help the seams from coming apart while handling. And when I quilt the quilt at the end, I change how I do that.....if I have a lot of little pieces in the quilt, I sometimes will do a denser quilting so as to help mail those seams down and make the quilt a bit more supported. I have had seams come apart after quilting and it is a bummer....but can be fixed either by hand stiching the seam back together or even putting a patch or a cute applique over it.

First of all Welcome to the QB from SE Michigan! I am sure there will be a great deal of response to this problem as this happens for a variety of reasons. I had this happen about a year ago on a baby quilt I made. Nothing like being frustrated! I had to remake the quilt using different fabrics. I too purchased the fabric from the same place you did. It was my first order from them. I knew that my seams were even 1/4" because narrower seams can cause this to happen too. Fabric quality is a bigger problem than the seams. I would strongly suggest pre-washing fabrics from this company maybe this will help too. Sorry this happened to you. Again welcome to the QB.

I would never wash a top alone. After the quilt is finished, I wash it. The seams stay in place because the quilt is quilted, either by machine or by hand, keeping the seams in tact. I only had a problem once and it was because I didn't watch my sewing when I did a QAYG sashing and my stitching ran off the material, thus not catching it all. I fabric glued it in place and kept the quilt for myself. :-) The glued part isn't showing but "I" know it's there. lol

Don't give up on the quilt just because of a few seams coming loose. If there's not enough fabric to hand stitch, then try putting a pretty applique over the bad area. Some really cute quilts have been made with kitty cats climbing all over them. cat silhouette applique patterns - Bing images You can use your zigzag stitch and reinforce the areas. We've all had to make some "extreme" decisions so that the project is saved.

I see that you said you ironed the seams open. That wouldn't make it fray, but I find I like how the batting doesn't show when I iron my seams to the side. When I first started quilting, I ironed my seams open and found I could see my sewing threads in between each part of the blocks. I didn't like that . The thread doesn't always match as good as I would like.

Without seeing the quilt and the frayed areas it's hard to tell why it happened. About 45 years ago I bought a patchwork kit from Woolworths. I think it cost $1.00 and was old looking, and they were clearing inventory because they were shutting down. I bought it to do by hand when I was traveling for work. Some of the fabric was questionable and that I didn't use - there was enough fabric to make a king size quilt and have a lot left over. It has held up okay, but then a few squares started to fray more because it was tied instead of quilted. (at the time I only tied my quilts.) I was lucky and was able to sew patches over some parts and then I did stitch in the ditch on the quilt and it has been fine. It's 35 years old. My daughter won't let me give it away. It's warm and huge.
Take some time and fix the frayed areas and use it. Each quilt is a learning experience. Be proud you finished it. Many quilts are started and never reach the finish line.