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I am completely new to the game and wanted to run this 1750 army through.

HQ

1 brother captain w/hammerhand
2 terminators
2 terminators w/ incinerators

Elite (for deep strike)

1 brother-captain
4 terminators

Troops

1 justicar
3 grey knights
1 grey knight w/incinerator

1 justicar
3 grey knights
1 grey knight w/incinerator

Fast attack (for deep strike)

1 justicar w/auspex
4 grey knights

Heavy Support

1 justicar
4 grey knights w/psycannons

1 justicar
4 grey knights w/incinerators

1 land raider crusader

I don't play GKs, but face them occasionally.

First thing that springs to mind is that you only have 2 small troop squads, meaning that your objective-holding ability can be nullified quite easily.
I'm not sure about the Fast Attack Gks, you could do with more troops.
I've never faced the Heavy Squads, so not sure if they are worth it.

The Land Raider could be bought as a Dedicated Transport for your Terminators, freeing up a HS slot.

One thing that Gks suffer from is poor/limited anti-tank. Most people seem to cover this with Land Raiders (bristling with LasCannon) and Dreadnoughts (again, lascannon variant).

I'm sure some more experienced GKer will come along and offer more/better advice soon

Land Raiders cannot be bought as dedicated transports for termies. Only Inquisitors get them dedicated. All three HS slots should be lascannon Land Raiders or dreadnaughts in lower point games, it is literally the only long range AT you have in the list unless you run SoB allies with multimeltas, even then you only have short range AT.

Never give PAGKs any special weapons because they lose their S6 in HtH for an overpriced gun. So take those purgation squads out, silly. Max out your termy squads with psycannons, 2 in the squad and one on the leader. Why? Because they only lose their crap storm bolter that they don't gain true grit with anyways.

Teleporting PAGKs also really blow, use the ally structure to get a SoB celestions with an Immolator if you really want a fast attack squad; other than that, GKs do not use fast attack except in absurdly large games.

Alright, example 1750 pure GK list:

HQ (In Land Raider)
Grand Master
-Frag grenades
-Master Crafted N.F.W.
-Psycannon
4xTerminators
-Psycannon

Troops
5x PAGKs
5x PAGKs
7x PAGKs
5xStormtroopers
-2xmeltas
5xStormtroopers
-2xmeltas

Heavy Support
Land Raider
2x Dreadnaught w/Lascannon & H.Flamer

Total 1750

Termies rush up and kill the biggest baddest dude the opponent has, using the GM's force weapon, making eternal warrior rules useless. Grey Knights stay back in objective missions, using their shrouding to full advantage, RMEMBER SHROUDING!!!!! (never seen someone use it more than a couple turns before forgetting altogether) Annihilation they stay just in storm bolter range to soften up stuff.

Dreadnaughts slowly advance, trying to pick off armor. Once in range of a unit of high armored dudes, he flames and assaults them; given that there are little armor left to do harm

The stormtroopers are your late game objective grabbers, and your close range anti-tank unit with meltas.

Lot of info, but there are few ways to play pure GKs effectively without LR spamming the board.

A little thing, a bit nit picky but if it's a PURE GK force... that means that there are ONLY Grey Knights, which means that that Stormtrooper squad is out of place for a Pure Grey Knight army.

Also, you cannot give two special weapons to Terminator Squads. Therefore at most you can have 2 Psycanons per squad, one on the Brother Captain and one on the squad. Also, avoid giving psychic powers to GKTs because usually it is a waste of points although Hammerhand is a good power, if you want to have it take Brother Captain Stern, who gets it automatically.

Here's a 1750 List.. not sure you'll like the lack of bodies.. but there is Heavy Support.

HQ:

Grand Master w/ Psycannon
4x GKTs w/ Psycannon

Troops:

3x (6 PAGK, 1 Justicar w/ Frags)

Heavy Support:

3x Land Raiders w/ Smoke Launchers

Short, sweet, to the point, and watch yor opponents face when you drop 3 Land Raiders on him. Put the PAGKs inside of the Land Raiders for objective capturing but as I said, stay back and shoot. Rinse and repeat as much as you can. I'm not huge fans of lists so heavily absent of bodies, but Grey Knights hammer first and ask questions later (And it works most of the time). Also a bit of strategy, Grey Knights are tactical monsters, we are a shoot first CC army only when necessary so don't go rushing into battle.

I think your initial list was preferable in its HQ selection. GKGM is too expensive. Yes, he's a combat monster, but he'll spend very little of the game in combat, so you're better off with the GKBC and more GKs.

Good advice from Matt1785 to not rush into battle. Stand back, move sideways, shoot shoot shoot. Only get into CC when you have to, then do it with overwhelming force.

3 Land Raiders is the tactically astute choice, but it won't make you any friends. 2 LRs and a dred with TLLC/ML is weaker but also saves points for more GKs. You need to make sure that your troops are capable of holding objectives.