Create a Report to Display Contact Birthdays in a Specific Month

The steps below will use the Relative Date Filter or Groupings. 

Create a report for birthdays in the next 30 days

1. Create a report on the Contact object
2. For the date filter, use "Created Date" with the range set as "All Time"
3. Add a custom filter for the "Birthdate" field, where Operator is equals, then select Use Relative date, and set to NEXT 30 DAYS
4. If desired, add another custom filter to select a specific account only

Create a report for birthdays in the current month

1. Create a report on the Contact object
2. For the date filter, use "Created Date" with the range set as "All Time"
3. Add a custom filter for the "Birthdate" field, where the Operator is equals, and Use Relative date is set to THIS MONTH

Create a report grouped by month

1. Create a report on the Contact object
2. For the date filter, use "Created Date" with the range set as "All Time"
3. If using the Classic Builder, change the report's format to "Summary"
4. Create a grouping on the "Birthdate" field
5. Select the arrow next to the new grouping and select Group Dates by | Calendar Month in Year