The nearest to Graz HA summit is HA/ND-001. It is on the HA/OE border, accessible with 30 min easy walk from Austrian side. There is a nice picnic site and look-out tower on the very summit. It might be busy on weekends, but there is enough place for even a full size dipole antenna.
Another near summit is HA/DD-016. It is almost a drive-on summit, you need only few minutes to walk. There are a concrete trig-tower (closed) and some small houses on the top. Place for antenna is limited. Site isn’t too attractive.
If you are ready to drive further into Hungary I’d suggest HA/KD-006 or HA/DD-012. Both are drive-on with look-out tower. Space for antenna is limited, and summit might be busy on weekends. DD-012 is on the southern bank of Balaton lake, so you can visit the beach after (or before) the activation. (WX forecast not really favorable for weekend)
If you have more time there are some really nice summits on the northern bank of lake Balaton (HA/KD-019, KD-022, KD-030, KD-041) with nice panorama, and look-out tower or ruins of medieval castles on the top. These summits are more challenging, requiring longer (1 - 2 hours) walk to reach.
If you need any further details about specific summit(s), just drop a message.