Is there a way to check the special moves in-game in Tony Hawk's Pro Skater?

Not in-game no. The very first game was pretty barebones compared to what it is today. There's no move listing, gap listing, trick editing... nothing. You had to find out what the tricks were yourself (per skater) or refer to a guide.

As mentioned by Jeff Mercado, there unfortunately isn't any way to check your special tricks in game. For reference, here is a complete list of special tricks for THPS1 on the PS1.

Skater:           Move Name:              Button Press:          Points: 

Bob Burnquist:    Backflip                ↑, ↓, ○                4000
(Vert)            Burntwist               ←, ↑ + ▲               5000
                  One Footed Smith        →, →, ▲                Varies

Kareem Campbell:  Frontflip               ↓, ↑, ○                4000
(Street)          Kickflip Underflip      ←, →, □                1500
                  Casper Slide            ↑, ↓, ▲                Varies

Rune Glifberg:    Kickflip McTwist        →, →, ○                4000 
(Vert)            Christ Air              ←, →, ○                2100 
                  Front-Back Kickflip     ↑, ↓, □                1575

Tony Hawk:        540 Board Varial        ←, ←, □                2000
(Vert)            360 Flip to Mute        ↓, →, □                1500
                  Kickflip McTwist        →, →, ○                4000
                  THE 900                 →, ↓, ○                8000!

Bucky Lasek:      Fingerflip Airwalk      ←, →, ○                2000   
(Vert)            Varial Heelflip Judo    ↓, ↑, □                2500
                  Kickflip McTwist        →, →, ○                4000

Chad Muska:       Frontflip               ↓, ↑, ○                4000
(Street)          360 Shove It-Rewind     →, →, □                1500
                  One Footed 5-0 Thumpin  →, ↓, ▲                Varies

Andrew Reynolds:  Backflip                ↑, ↓, ○                4000   
(Street)          Triple Kickflip         ←, ←, □                1500   
                  Heelflip to Bluntslide  ↓, ↓, ▲                Varies

Geoff Rowley:     Backflip                ↑, ↓, ○                4000
(Street)          Double Hardflip         →, ↓, □                1500
                  Darkslide               ←, →, ▲                Varies

Elissa Steamer:   Backflip                ↑, ↓, ○                4000
(Street)          Judo Madonna            ←, ↓, ○                1500
                  Primo Grind             ←, ←, ▲                Varies

Jaime Thomas:     Frontflip               ↓, ↑, ○                4000
(Street)          540 Flip                ←, ↓, □                1500   
                  One Footed Nosegrind    ↑, ↑, ▲                Varies

Officer Dick:     Yeehaw Frontflip        ↓, ↑, ○                4000
(Vert/Street)     Assume the Position     ←, ←, ○                1575
(READ BELOW!)     Neckbreak Grind         ←, →, ▲                Varies

Private Carrera:  The Well Hardflip       →, ←, □                5500
(Vert/Street)     Somi Spin               ←, ↓, ○                5500
                  Ho Ho Ho                ←, ↑, ▲                2000

Adapted from Slateman's FAQ/Move List

Interestingly, while you can't look up any of the tricks in-game, there was an incomplete list of special tricks in the game's instruction manual.