No, the flu vaccine does not trigger asthma attacks. In fact, it is safe for people with asthma to receive the flu vaccine2. The nasal spray vaccine (live attenuated influenza vaccine) can be used for under 18 year olds if they are not currently experiencing an acute exacerbation of symptoms including increased wheezing and/or needed additional bronchodilator treatment in the previous 72 hours4.
2 Flu Vaccinations | Asthma + Lung UK (asthmaandlung.org.uk)
4 Child Flu Vaccine - NHS (www.nhs.uk)