Is the word grama the same in Brazilian and European Portuguese?

No

In Brazilian Portuguese, grama refers to the green vegetation covering the ground (grass). In Continental Portuguese, the word grama is almost exclusively used to refer to the unit of mass (grams). To refer to the green vegetation in a garden or park, a person from Portugal would use the words relva or erva.

Brazilian Portuguese Examples:

1. Eu preciso cortar a grama hoje. (I need to mow the grass today.)
2. A grama do jardim está muito verde. (The garden grass is very green.)
3. O cachorro adora correr na grama. (The dog loves to run on the grass.)
4. Cuidado para não pisar na grama molhada. (Be careful not to step on the wet grass.)
5. A grama secou por causa do calor. (The grass dried up because of the heat.)

Continental Portuguese Examples:

1. Eu preciso de cortar a relva hoje. (I need to mow the grass/lawn today.)
2. A relva do jardim está muito verde. (The garden grass is very green.)
3. O cão adora correr na relva. (The dog loves to run on the grass.)
4. Cuidado para não pisares na relva molhada. (Be careful not to step on the wet grass.)
5. A relva secou por causa do calor. (The grass dried up because of the heat.)