Refer below image:

Formula in cell D2 - 1st Qtr (Feb to Apr):
=SUMPRODUCT((MONTH(1&LEFT(A2:A100,3))>=2)*(MONTH(1&LEFT(A2:A100,3))<=4)*(B2:B100))
Formula in cell D3 - 2nd Qtr:
=SUMPRODUCT((MONTH(1&LEFT(A2:A100,3))>=5)*(MONTH(1&LEFT(A2:A100,3))<=7)*(B2:B100))
Formula in cell D4- 3rd Qtr:
=SUMPRODUCT((MONTH(1&LEFT(A2:A100,3))>=8)*(MONTH(1&LEFT(A2:A100,3))<=10)*(B2:B100))
Formulas in cell D5 & D6 (2 options) - 4th Qtr (Nov, Dec & Jan):
=SUMPRODUCT((MONTH(1&LEFT(A2:A100,3))>=11)*(B2:B100)+(MONTH(1&LEFT(A2:A100,3))=1)*(B2:B100))
=SUMPRODUCT(((MONTH(1&LEFT(A2:A100,3))>=11)+(MONTH(1&LEFT(A2:A100,3))=1))*(B2:B100))
Formula in cell C2 - 1st Half - Feb to July:
=SUMPRODUCT((MONTH(1&LEFT(A2:A100,3))>=2)*(MONTH(1&LEFT(A2:A100,3))<=7)*(B2:B100))
Formula in cell C3 - 2nd Half:
=SUMPRODUCT(((MONTH(1&LEFT(A2:A100,3))>=8)+(MONTH(1&LEFT(A2:A100,3))=1))*(B2:B100))
These formulas will be flexible and work without a helper column.
You may also shorten the formulas and use as below
Ex. for 1st Qtr (Feb to Apr):
=SUMPRODUCT((MONTH(1&LEFT(A2:A100,3))={2,3,4})*(B2:B100))
For 2nd Qtr:
=SUMPRODUCT((MONTH(1&LEFT(A2:A100,3))={5,6,7})*(B2:B100))
For 3rd Qtr:
=SUMPRODUCT((MONTH(1&LEFT(A2:A100,3))={8,9,10})*(B2:B100))
and for 4th Qtr (Nov, Dec & Jan):
=SUMPRODUCT((MONTH(1&LEFT(A2:A100,3))={1,11,12})*(B2:B100))
and similarly for 6-mths.
Regards,
Amit Tandon
www.globaliconnect.com