I don't think there's a 'core' QGIS function for this. However, install the MMQGIS plugin, and use the 'hub distance' tool. Chose your points as the "Source Points" layer, and your lines as the "Destination Hubs Layer" and set "Output Shape Type" to "Line to Hub." This will give you a shapefile of lines (with length) that join the closest features. Conveniently, it also adds the ID of the closest feature to the attribute table.
Edit/Comment: This approach works well if there are fewer points than lines, or else a one-to-one line-to-point situation. Having more points than lines complicates things. 'Hub Distance' tool adds the ID of the line feature to the point attribute table as 'HubName' along with the distance between the point and the center of the line as 'HubDist'. If there are more points than lines, additional work is required to determine within each subset of points (those multiple records linked to each line hub) which of those is the closest to each line, and/or which is one with the desired set of attributes to join to the line.