¿Cómo crear una vista que muestre los datos de los últimos 4 trimestres?

Mmm. Si interpreto "los últimos cuatro trimestres" como "los últimos cuatro trimestres que están en los datos", entonces aquí hay un método.

Primero, puedes obtener los cuatro cuartos haciendo:

select distinct p_year, p_quarter
from production p
order by p_year desc, p_quarter desc
limit 4;

Puede enumerarlos usando variables:

select p_year, p_quarter, (@rn := @rn + 1) as enum
from (select distinct p_year, p_quarter
      from production p
      order by p_year desc, p_quarter desc
      limit 4
     ) p cross join
     (select @rn := 0) params

Luego, puede usar esto en una consulta para pivotar los datos:

select p.person_id,
       sum(case when seqnum = 1 then num else 0 end) as num_quarter_1,
       sum(case when seqnum = 2 then num else 0 end) as num_quarter_2,
       sum(case when seqnum = 3 then num else 0 end) as num_quarter_3,
       sum(case when seqnum = 4 then num else 0 end) as num_quarter_4
from production p join
     (select p_year, p_quarter, (@rn := @rn + 1) as seqnum
      from (select distinct p_year, p_quarter
            from production p
            order by p_year desc, p_quarter desc
            limit 4
           ) p cross join
           (select @rn := 0) params
     ) yq
     using (p_year, p_quarter)
group by p.person_id;

EDITAR:

Si está definiendo los trimestres en función del trimestre actual, puede hacer algo bastante similar:

select p.person_id,
       sum(case when seqnum = 1 then num else 0 end) as num_quarter_1,
       sum(case when seqnum = 2 then num else 0 end) as num_quarter_2,
       sum(case when seqnum = 3 then num else 0 end) as num_quarter_3,
       sum(case when seqnum = 4 then num else 0 end) as num_quarter_4
from production p join
     (select year(curdate()) as p_year, quarter(curdate()) as p_quarter, 1 as seqnum union all
      select year(curdate() - interval 1 quarter) as p_year, month(curdate() - interval 1 quarter) as p_quarter, 2 as seqnum union all
      select year(curdate() - interval 2 quarter) as p_year, month(curdate() - interval 2 quarter) as p_quarter, 3 as seqnum union all
      select year(curdate() - interval 2 quarter) as p_year, month(curdate() - interval 3 quarter) as p_quarter, 4 as seqnum           
     ) yq
     using (p_year, p_quarter)
group by p.person_id;

Hay otros enfoques, como:

select person_id,
       sum(case when year(curdate()) = p_year and quarter(curdate()) = p_quarter
                then num else 0
           end) as num_quarter_1,
       sum(case when year(curdate() - interval 1 quarter) = p_year and quarter(curdate() - interval 1 quarter) = p_quarter
                then num else 0
           end) as num_quarter_2,
       sum(case when year(curdate() - interval 2 quarter) = p_year and quarter(curdate() - interval 2 quarter) = p_quarter
                then num else 0
           end) as num_quarter_3,
       sum(case when year(curdate() - interval 3 quarter) = p_year and quarter(curdate() - interval 3 quarter) = p_quarter
                then num else 0
           end) as num_quarter_4
from production p
group by person_id;