Puede usar lo siguiente, que usa una función agregada con un CASE declaración:
select p.name,
v.v_name,
sum(case when Month(py.pay_date) = 4 then amount end) april_amount,
sum(case when Month(py.pay_date) = Month(curdate())
then amount end) current_month_amount,
sum(case when Month(py.pay_date) = Month(curdate())-1
then amount end) previous_month_amount
from persons p
left join vehicle v
on p.id = v.person_veh
left join payment py
on p.id = py.person_id
group by p.name,
v.v_name
