You certainly don't need a sparky for that. Look at the cable and see if it says RG6 or RG59 on the jacket. Go down to the local HD, Lowe's, or hardware and buy a small pack of male twist-on RG6 or RG59 ends( you'll only need 2, but they come in packs of 5). On the pack it will diagram how to strip the cable to prepare for the ends. You'll also need to buy the appropriate coupler. This is a barrel shaped device with females on both ends that the new cable ends will screw into. That's it!

FWIW, the "twist on" cable fitting is barely better than just twisting the center conductors together. There are several things involved in a proper repair of a "dog chew".

1. make sure it won't happen again, by whatever method seems appropriate.

1(A) eliminate the dog, via any suitable method.
1(B) re-route the cable so as to be out of the dog's reach.
1(C) provide remedial training for the dog, to discourage future episodes.

2. Inspect the cable for other damage. If there are teeth marks all over it for several feet, consider replacement of the entire run or a larger portion thereof.

3. Use quality fittings and weatherproofing. Twist-on fittings are at the bottom of the barrel, and are not worth the time to install. Two piece crimp-ons are the same. One piece hex crimp fittings are OK, with "Snap-n-seal" or similar commercial grade fittings at the top.

4. Use a piece of heatshrink tubing over exposed splices if possible.

Alternate method- slip the cable man/woman a $20 and the problem will usually go away in a few minutes with minimal effort on your part.

I assumed this dog attack on the co-ax was indoors. If so, twist on connectors are fine. If it's outside as wayne440 suggests, then follow his instruction #3.

First you will need someone to hold him down, especially if he is a big dog. Then you will need a large needle, some cat gut, and a pot of boiling water.

Oh, splice the cable? Nevermind.

I had a springer spaniel puppy that chewed up the bedroom clock radio ac cord. Grew up and still chewed on it. It liked that spot behind the headboard. Maybe just for the chew toy.
Don't know why that dogs hair never stood on end.