I dont believe block group data were added to the online tool until the 2009-2013 american Community Survey (ACS) 5-year estimates. For 2010 data, I assume you want the 2006-2010 ACS 5-year estimates. The 2010 Decennial Census did not ask questions about income, thus you need to get the data from the 2006-10 ACS 5-year estimates.
To get the income data at block group level, you will need to use the Summary Files. See the documentation on the Census website here:
https://www.census.gov/programs-surveys/acs/technical-documentation/summary-file-documentation.2010.html
If you only need the one data table you can use the Excel lookup tool they have. Or, better yet, check out the IPUMS National Historical Geographic Information System (NHGIS) project where you can get the data for any year plus associated GIS files here:
https://www.nhgis.org/