Incrementar o valor da coluna em determinada condição na consulta SQL no Postgresql

Use lag() e soma acumulada:

with query as (
select w.*,
       sum(case when week < prev_week + 2 then 0 else 1 end) over (partition by animal order by week) as grp
from (select w.*,
             lag(week) over (partition by animal order by week) as prev_week
      from test.walk w
     ) w
)
select
    animal, week, grp,
    dense_rank() over (order by animal, grp) as grp2
from query

Nota:Isso reiniciará a contagem para cada animal - o que parece ser a intenção real do que você deseja fazer. A definição do problema é um pouco complicada se você quiser que os grupos sejam separados por animal, mas que sejam incrementais. Um método é:

select w.*,
       sum(case when prev_week = week then 0 else 1 end) over (order by min_week, animal, week) as grp
from (select w.*,
             lag(week) over (partition by animal order by week) as prev_week,
             min(week) over (partition by animal) as min_week
      from test.walk w
     ) w;