Using a combination of the techniques provided by others I was able to return the values for both our company's fiscal quarter and fiscal year, which in our case starts in July. First section uses choose to map the month to the corresponding FY Quarter, and second section checks to see if the month is before or after July to get it into the right Fiscal year as well.
A date of 7/5/2019 ends up coming out as "Q1 FY20"
=IF(ISBLANK([Date]),"",(CONCATENATE("Q",(CHOOSE(MONTH([Date]),3,3,3,4,4,4,1,1,1,2,2,2))," FY",(IF(MONTH([Date])>=7,RIGHT(YEAR([Date])+1,2),RIGHT(YEAR([Date]),2))))))
Original poster's question would use the following for April:
=IF(ISBLANK([Date]),"",(CONCATENATE("Q",(CHOOSE(MONTH([Date]),4,4,4,1,1,1,2,2,2,3,3,3))," FY",(IF(MONTH([Date])>=4,RIGHT(YEAR([Date])+1,2),RIGHT(YEAR([Date]),2))))))