This is a classic Gaps-and-Islands problem.

You first need to aggregate up the data per year, and check for the total count per year.

Then use gaps-and-islands techniques to split it into groups of rows which are consecutive years, finally taking the min and max dates for each of those groups.
In older versions of SQL Server, use YEAR( instead of DATETRUNC(year, , the latter is more performant if available.

WITH Data AS (
    SELECT
      t2.ID,
      Year = DATETRUNC(year, t2.DAT),
      MinDate = MIN(t2.DAT),
      Maxdate = MAX(t2.DAT),
      PrevYear =   LAG(DATETRUNC(year, t2.DAT)) OVER (PARTITION BY t2.ID ORDER BY DATETRUNC(year, t2.DAT)),
      NextYear =  LEAD(DATETRUNC(year, t2.DAT)) OVER (PARTITION BY t2.ID ORDER BY DATETRUNC(year, t2.DAT))
    FROM Table1 t1
    JOIN Table2 t2
      ON t2.ID = t1.ID
     AND t2.DAT >= DATEADD(day, -300, t1.INDX)
    GROUP BY
      t2.ID,
      DATETRUNC(year, t2.DAT)
),
Islands AS (
    SELECT
      *,
      NextMaxDate = LEAD(d.MaxDate) OVER (PARTITION BY d.ID ORDER BY d.Year)
    FROM Data d
    WHERE d.PrevYear IS NULL
       OR d.PrevYear < DATEADD(year, -1, Year)
       OR d.NextYear IS NULL
       OR d.NextYear > DATEADD(year, 1, Year)
)
SELECT
  ID,
  Start = MinDate,
  [End] = ISNULL(NextMaxDate, MaxDate)
FROM Islands i
WHERE i.PrevYear IS NULL
   OR i.PrevYear < DATEADD(year, -1, Year);

db<>fiddle

I'm not sure I got the proper filters you wanted around the dates so adjust as necessary. The main portion of the logic uses universally available date functions and there's no need to use the value in qrtr at all:

with data as (
    select t1.indx, t2.id, t2.dat,
        datediff(year, min(dat) over (partition by t2.id), t2.dat) -
        dense_rank() over (partition by t2.id order by year(t2.dat)) as grp
    from table1 t1 left outer join table2 t2 on t2.id = t1.id
    where t2.code = 'A' and t2.dat >= dateadd(day, -300, t1.indx)
)
select id, min(dat) as min_dat, max(dat) as max_dat
from data
group by id, grp
order by id, min(dat);