Recently installed solid blanket of St. Augustine grass and has done very well for 6 weeks with frequent watering but am beginning to see a few brown spots and want to know what to fertilize with in order to build a strong root system

Brown spots may be the result of too much water. In this case, horticultural cornmeal applied at 20 pounds per 1000 square feet should help. It certainly would not hurt.
Have you had your soil tested? A great kick start fertilizer is corn gluten meal. I prefer the powder (non-granulated) form. If it is windy, do not attempt to apply the powdered form. You will end up losing a good bit of it and you will end up looking like something out of a science fiction movie! Did you put compost between the squares of sod?

Fertilizer is nice but to build a strong root system, you need to water properly. Sounds like you're still watering on the sod installer's recommendation. You need to move to the "regular" routine of once per week.

See how long you can go without watering at all before your grass wilts. Then water it for an hour or until you get runoff (which ever is sooner). Then go again until the grass wilts. If it wilts within a week, you need to water longer next time. If it goes a week without wilting, then your roots are getting down where they belong.

You can help the roots by letting the grass get longer, too. If you mow at 3-4 inches, rather than 1-2 inches, your roots will grow deeper and you'll have much less in the way of weed problems.

The last thing you can do for your grass is fertilize, but the first two items above are much more important. Fertilize with a good organic fertilizer. The one I've seen used with unbelievable results is Texas Tee. I don't have the wallet for that so I use corn meal, alfalfa pellets, and any other ground up grain I can find cheap at feed stores. The application rate for whatever you find is 10-20 pounds per 1,000 square feet.

Usually the problem with brown spots on new sod St Augustine lawns is that they were initially over fertilized at the sod farm. First thing you need is corn meal, then worry about fertilizing. Give the corn meal 3 weeds to show good results.