To know the current compatibility_level of your database you can run:

SELECT name,compatibility_level 
FROM sys.databases 
WHERE name = 'DatabaseName';

Result for my TEST database:

name    compatibility_level
TEST    140

If the compatibility level changes, for example with the command:

ALTER DATABASE TEST SET COMPATIBILITY_LEVEL = 120;

You could find the date and time that happened in the sql server error log:

Or reading the current error log file with xp_readerrorlog:

EXEC sp_readerrorlog 0, 1, 'COMPATIBILITY_LEVEL' 

Result

To know the current compatibility_level of your database you can run:

SELECT name,compatibility_level 
FROM sys.databases 
WHERE name = 'DatabaseName';

Result for my TEST database:

name    compatibility_level
TEST    140

If the compatibility level changes, for example with the command:

ALTER DATABASE TEST SET COMPATIBILITY_LEVEL = 120;

You could find the date and time that happened in the sql server error log:

Or reading the (current) error log file with xp_readerrorlog:

EXEC sp_readerrorlog 0, 1, 'COMPATIBILITY_LEVEL' 

Result

Replace the first parameter of xp_readerrorlog to 1 to read the second error log file, 2 for the third file, ... .