Where might this be, but Beaver Brook in Belmont. It's just a hop, skip and a jump from Waverly. Which you can get to by train from North Station or by bus from Harvard Square.
The southern historic section is the most easily accessible from Waverly, and where you will find most of the fun toddler activities. Once out of the train or bus stop, walk northwest along Trapello Road. About 200yds you'll reach the playground on your right, a little further on your left up Mill Street is Duck Pond and nice wooded trails.
If you are looking for more trail walking go to further north to the Rock Meadow section that extends into Waltham. If you need still more trails to walk along, just keep walking along the Western Greenway, walk to Belmont Center, downtown Waltham, or all the way around back to Waverly. But I digress, this degree of walking is mosty for older kids.
First just take your toddler out on the wooded trails and paths. Find sticks to bang against rocks, fish & bugs to watch, and dogs to pet. That is fun enough.