Sorting chart chronologically not alphabetically

Two ideas. Both of them require you to re-do your data:

1) Use actual dates instead of text strings to define your quarters, such as using the first day of the quarter (Date format in this example is US)

Quarter	Q Date
Apr - Jun 18	4/1/2018
Apr - Jun 18	4/1/2018
Jul - Sep 18	7/1/2018
Jul - Sep 18	7/1/2018
Oct - Dec 18	10/1/2018

2) Use a string that sorts properly:

Quarter	Q String
Apr - Jun 18	2018 Q2
Apr - Jun 18	2018 Q2
Jul - Sep 18	2018 Q3
Jul - Sep 18	2018 Q3
Oct - Dec 18	2018 Q4

Have a look at the attached file.
As per above, you need actual dates to sort by date.
One one is using some helper columns to derive a date from the text strings. then add these fields to your pivot table and sort accordingly.
You then could 'hide' the helper columns if required.

Formula:

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[@Quarter] is a reference to the Quarter column in the table. This is called a structured reference (which you can google for more details). In this case it mean "the value in column Quarter that is in the same row as this formula".

The format of VLOOKUP is (value, range, column[, approximate])

value is the value to search for in the first column of range.
column is the relative number of the column in range to retrieve the result from the same row where value was found.

approximate is optional and defaults to TRUE. If it is FALSE then VLOOKUP will only look for an exact match. If it is TRUE and the data is sorted in ascending order, if VLOOKUP cannot find an exact match it will find the value closest to value without going over it. ("1" should not be used here since a boolean value is expected. In this case Excel will interpret "1" as FALSE but that is a bit convoluted when you can just use FALSE to begin with.)