I don't know about Gompf and Stipciz, but the algorithm to obtain a Heegaard diagram from a surgery presentation may be found for example in A Simple Proof of the Fundamental Theorem of Kirby Calculus on Links by Ning Lu, Trans. Amer. Math. Soc. Vol 331(1) pp. 143-156 (1992). It's not difficult, and it's worth knowing.

I don't know about Gompf and Stipsicz, but the algorithm to obtain a Heegaard diagram from a surgery presentation may be found for example in A Simple Proof of the Fundamental Theorem of Kirby Calculus on Links by Ning Lu, Trans. Amer. Math. Soc. Vol 331(1) pp. 143-156 (1992). It's not difficult, and it's worth knowing.