Are bronchial spasms related to allergies?

Yes. Allergies are one of the predominant cause of bronchial spasms. Other causes include exposure to cold air, irritants (cigarette or other smoke, strong perfumes, chemicals), infections, exercise and poor air quality.
Asthma. Classic asthma has spasm (bronchial hyper-reactivity) at its core and 70% of asthma symptoms are triggered by allergens or allergies. Not all asthma is allergy-related but a rather large portion.