No, both of these poems are not sonnets. First of all, both poems are more than fourteen lines long. Second of all, they do not follow the regular rhyme pattern of either Italian or English sonnets.
The Passinate Shepherd to his Love is a pastoral poem, which means relating to or being a literary or other artistic work that portrays or evokes rural life, usually in an idealized way. To sum this poem up, it's a shepherd who's looking for a lady love and using his surroundings as a way to entice her.
The other poem, The Nymph's Reply to the Shepherd, is a parody. It mimics and responds from a female perspective the first poem. The narrator in the poem is refusing the shepherd's request.