There doesn't seem to be any dedicated LSM to PNG converter available. However, as LSM images are based on TIFF, you might want to try to rename LSM extension to TIF and then use some TIF to PNG converter. Could work this way.
Convert Ziess LSM microscope image to Portable Network Graphic.
We have found one software suitable in our database for this conversion.
Multiplatform software - convert lsm to png on any platform
Software tool for reading and writing image data using standardized, open formats
Convert from lsm file format
The table with program actions contains information about what each program is capable of doing with their files. This may be a good pointer for further searches for certain file conversions and help you find the converter you are looking for if our search result is insufficient for you. However, it is far from perfect and may sometimes show results which are not really usable because of the nature of the software handles the files. So a conversion that appears as possible, just because of matched actions, may in fact not be possible.