Can you have an allergic reaction to paprika if you have tree nut allergy?

Yes - according to a paper published, there is significant cross-reactivity between chili peppers and Brazil nut and hazelnut at least in laboratory assays, however clinical reactivity was not correlated in this paper (Ivens et al., PMID 34714660):

"The xMAP food allergen detection assay (xMAP FADA) ...

Patients with tree nut allergies do not typically react to paprika. Paprika allergy is primarily associated with the celery-birch-mugwort-spice syndrome, where cross-reactivity occurs through pathogenesis-related proteins (PR-10), and profilins shared with birch and mugwort pollen, rather than throu...