If you want to copy a blob from one storage container to other storage container, you could use beginCopy method, firstly get the source blob url with getBlobUrl method then pass it.
If you want a sample you could refer to this github sample:BlobAsyncClientBaseJavaDocCodeSnippets.
And if you want to move one blob from source container to another container and it doesn't exist in the source container, for now no direct method to implement, you could copy the blob firstly, after copy activity then delete the source blob with delete method.
Actually from all these method link you could find they all provide the github sample just follow the project structure.
Update: if you want a sample code, you could refer to my below code, I have test it it could work.
String connectStr = "storage account connection string";
BlobServiceClient blobServiceClient = new BlobServiceClientBuilder().connectionString(connectStr).buildClient();
BlobContainerClient containerClient = blobServiceClient.getBlobContainerClient("test");
BlobContainerClient destcontainer=blobServiceClient.getBlobContainerClient("testcontainer");
PagedIterable<BlobItem> blobs= containerClient.listBlobs();
for (BlobItem blobItem : blobs) {
System.out.println("This is the blob name: " + blobItem.getName());
BlobClient blobClient=containerClient.getBlobClient(blobItem.getName());
BlobServiceSasSignatureValues sas = new BlobServiceSasSignatureValues(OffsetDateTime.now().plusHours(1),
BlobContainerSasPermission.parse("r"));
String sasToken = blobClient.generateSas(sas);
BlobClient destblobclient=destcontainer.getBlobClient(blobItem.getName());
destblobclient.beginCopy(blobClient.getBlobUrl()+ "?" + sasToken,null);
}

Update:
String connectStr = "source storage account connection string";
String destconnectStr="destination storage account connection string";
BlobServiceClient blobServiceClient = new BlobServiceClientBuilder().connectionString(connectStr).buildClient();
BlobServiceClient destblobServiceClient = new BlobServiceClientBuilder().connectionString(destconnectStr).buildClient();
BlobContainerClient containerClient = blobServiceClient.getBlobContainerClient("test");
BlobContainerClient destcontainer=destblobServiceClient.getBlobContainerClient("destcontainer");
PagedIterable<BlobItem> blobs= containerClient.listBlobs();
for (BlobItem blobItem : blobs) {
System.out.println("This is the blob name: " + blobItem.getName());
BlobClient blobClient=containerClient.getBlobClient(blobItem.getName());
BlobClient destblobclient=destcontainer.getBlobClient(blobItem.getName());
destblobclient.beginCopy(blobClient.getBlobUrl(),null);
}