A simpler way to check for the presence of hydrogen bonding is to compare the acidity of salicylide with its para isomer.
Case 1: No hydrogen bonding
Let us assume that there is no hydrogen bonding for the following analysis.
The $\ce{-CHO}$ group is an electron withdrawing group ($-M$ effect). The effect of acidity due to mesomeric effect at both ortho and para positions are identical. Therefore, we can ignore the effect of mesomeric effect for our analysis.
However, the $\ce{-CHO}$ group is slightly electron withdrawing group due to inductive effect. The strength of inductive effect varies with distance: farther away you are, the lesser is the effect. As the aldehyde in ortho position is closer to the carbinol carbon, effect of inductive effect on the carbinol carbon is more pronounced when compared to the case where the aldehyde is in the para position. As a result, the positive charge developed on the carbinol carbon is higher when the aldehyde is at ortho position. Therefore, the acidity of salicylaldehyde must be higher than that of its para isomer.
If there was no hydrogen bonding, then we should expect the acidity of salicylaldehyde to be higher than its para isomer.
Case 2: Hydrogen bonding
Salicylaldehyde forms intramolecular hydrogen bonding and its para isomer forms intermolecular hydrogen bonding. The effect of acidity due to intermolecular hydrogen bonding is lesser than the effect of hydrogen bonding due to intramolecular hydrogen bonding. This is because intermolecular hydrogen bonds can be broken easily as they are between two molecules and the molecules are in thermal motion.
If the effect of difference in hydrogen bonding in the two isomers is stronger than the effect of difference in inductive effect shown by the aldehydic group in the two isomers, we should expect a reversal of acidity order of that obtained in the previous case, i.e: salicylaldehyde is a weaker acid than its para isomer.
Experimentally measured values:
Molecule pKa
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Salicylaldehyde 8.37
4-hydroxybenzaldehyde 7.61
As the measured values suggest, intramolecular hydrogen bonding exists in salicylaldehyde.